LMIs in Control/pages/Delay Dependent Time-Delay Stabilization

Stabilization of Time-Delay Systems - Delay Dependent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system while being dependent on the delay.

The SystemEdit

For this particular problem, suppose that we were given the time-delayed system in the form of:

 

where

 

Then the LMI for determining the Time-Delay System for the Delay-Dependent case would be obtained as described below.

The DataEdit

In order to obtain the LMI, we need the following 3 matrices:  .

The Optimization ProblemEdit

Suppose - for the time-delayed system given above - we were asked to design a memoryless feedback control law where   such that the closed-loop system:

 

is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

The LMI: The Delay-Dependent Stabilization of Time-Delay SystemsEdit

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a scalar  , a symmetric matrix   and a matrix   that satisfy the following:

 

where 

Conclusion:Edit

Given the resulting stabilizing control gain matrix  , it can be observed that the matrix is asymptotically stable from the time-delay system where this gain matrix was derived.

ImplementationEdit

  • Example Code - A GitHub link that contains code (titled "DelayDependentTimeDelay.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIsEdit

External LinksEdit

A list of references documenting and validating the LMI.

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